A276022 n^2 * a(n) = 3*(5*n^2 - 5*n + 2) * a(n-1) - 16*(5*n^2 - 10*n + 6) * a(n-2) + 36*(5*n^2 - 15*n + 12) * a(n-3) - 144*(n-2)^2 * a(n-4), with a(0)=1, a(1)=6, a(2)=30, a(3)=144.

1, 6, 30, 144, 690, 3348, 16536, 83232, 426618, 2223180, 11756052, 62959680, 340881792, 1862954784, 10262937600, 56926831104, 317632207194, 1781352834300, 10034760283356, 56748881420640, 322033934657628, 1833043230774360, 10462349697348528, 59861990921495616

Offset: 0

Author

Gheorghe Coserea, Aug 23 2016

Keywords

• nonn

Examples

			A(x) = 1 + 6*x + 30*x^2 + 144*x^3 + ... is the g.f.
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..301
• Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Crossrefs

Cf. A091401, A276018.

Programs

Magma

I:=[6,30,144,690]; [1] cat [n le 4 select I[n] else (3*(5*n^2-5*n+2)*Self(n-1)-16*(5*n^2-10*n+6)*Self(n-2)+36*(5*n^2-15*n+12)*Self(n-3)-144*(n-2)^2*Self(n-4)) div n^2: n in [1..30]]; // Vincenzo Librandi, Aug 25 2016

Mathematica

a[0] = 1; a[1] = 6; a[2] = 30; a[3] = 144; a[n_] := a[n] = (3(5n^2 - 5n + 2) a[n-1] - 16(5n^2 - 10n + 6)a[n-2] + 36(5n^2 - 15n + 12) a[n-3] - 144(n-2)^2 a[n-4])/n^2;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 19 2018 *)

PARI

seq(N) = {
  my(a = vector(N)); a[1] = 6; a[2] = 30; a[3] = 144; a[4] = 690;
  for (n=5, N,
       my(t1 = 3*(5*n^2 - 5*n + 2)*a[n-1] - 16*(5*n^2 - 10*n + 6)*a[n-2],
          t2 = 36*(5*n^2 - 15*n + 12)*a[n-3] - 144*(n-2)^2 * a[n-4]);
       a[n] = (t1+t2)/n^2);
  concat(1,a);
};
seq(25)

Formula

n^2 * a(n) = 3*(5*n^2 - 5*n + 2) * a(n-1) - 16*(5*n^2 - 10*n + 6) * a(n-2) + 36*(5*n^2 - 15*n + 12) * a(n-3) - 144*(n-2)^2 * a(n-4), with a(0)=1, a(1)=6, a(2)=30, a(3)=144.

0 = x*(x+2)*(x+3)*(x+4)*(x+6)*y'' + (5*x^4 + 60*x^3 + 240*x^2 + 360*x + 144)*y' + 4*(x^2+6*x+6)*(x+3)*y, where y(x) = A(x/-12).

a(n) ~ 2^n * 3^(n+3/2) / (Pi*n). - Vaclav Kotesovec, Aug 25 2016